368 XV NORMED ALGEBRAS AND SPECTRAL THEORY constant y > 0 such that (15.8.14.1) V(x\y) = (Ufa) \Ufry) (for the scalar product defined in (15.7.4)). If A is finite-dimensional, then the image of A under Ul is the algebra Endc(I) of all endomorphisms of the vector space I, and the relation (15.8.14.1) remains valid, for the scalar product defined in (15.7.4.1) (from the scalar product on I -which is the restriction of that on A). We can assume that A is the Hilbert sum of a (finite or infinite) sequence of minimal left ideals !„ = Aen, where I = Ix (15.8.11.1) and all the In are isomor- phic (15.8.13). If x T£ 0 and Ufa) = 0, that is, if xl = {0}, we should have (Ax)l = {0}, and since the ideal Ax is nonzero, it contains a minimal left ideal I7 (15.8.8), which must be isomorphic to I (15.8.13); hence I'l = {0}, contrary to (15.8.12(iii)). Hence the representation £/r is faithful. Put Pn = t/i(en), which is the orthogonal projection of I = Ael onto the one-dimensional subspace en A*i (15.8.12), because (xe1 - enxel \ enyej = (enxei - el xel \ye^ = 0. Since emen = 0 if m ^ n, we have PmPn=0 and therefore the subspaces enAei are orthogonal in pairs. Moreover, I is the Hilbert sum of the sub- spaces enAel9 because, if xe1 is orthogonal to all these subspaces, we have Pn(xe1) = 0 for all n, so that enxe± = 0 for all «, and therefore xe1 belongs to the right annihilator of A, which is zero (15.7.5.7). This shows that the sequence (ln) is finite if and only if I (and therefore each In) is finite-dimQn- sional over C, or equivalently if and only if A is finite-dimensional. Let (an) be an orthonormal basis of I such that an e en Ael for each n. Then anafeenAen, hence ana* = !„<?„ for some AB e C* (15.8.12). Likewise We have ^ = An, because on the one hand and on the other hand «!• anan a* = % *n el<*n = *>'n ^n * * = ^ An en , since an et = an. Also 1 = fa, I flj = (a,, I en an) = (ana* \ en) = Xn(en \ en) and 1 = (fl. IO = (^ I a, cx) = (a*aM | ej = An(^ | ej; hence (15.8.14.2) •feU) = fc|e,)ce also contains